Mathematical tools for intermediate economics classes
Iftekher Hossain
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Calculus of Multivariable Functions

Section 2

Partial derivatives and the rules of differentiation

Practise questions

1. Find the first-order partial derivatives for each of the following functions:

  1. \(f(x,y) = 6x^{2} + 8y^{2}\)
  2. \(f(x,y) = 6x^{2} + 10xy + 8y^{2}\)
  3. \(z = 6x^{2} + 10xy + 8y^{2} + 100\)
  1. \(f_{x} = 12x \qquad\qquad\qquad\quad f_{y} = 16y\)
  2. \(f_{x} = 12x + 10y \qquad\qquad f_{y} = 10x + 16y\)
  3. \(z_{x} = 12x + 10y \qquad\qquad z_{y} = 10x + 16y\)

2. Use the product rule to find the first-order partial derivatives for each of the following functions:

  1. \(f(x,y) = x^{2}(3x + 2y)\)
  2. \(f(x,y) = (x + y)(x - y)\)
  1. \(f_{x} = (3x + 2y)\frac{\partial }{\partial x}(x^{2}) + x^{2}\frac{\partial }{\partial x}(3x + 2y)\)

    \(\qquad = (3x + 2y)(2x) + x^{2}(3) = 9x^{2} + 4xy\)

      And, \(f_{y} = (3x + 2y)\frac{\partial }{\partial y}(x^{2}) + x^{2}\frac{\partial }{\partial y}(3x + 2y)\)

      \(\qquad\quad\; = (3x + 2y)(0) + x^{2}(2) = 2x^{2}\)

  2. \(f_{x} = (x + y)\frac{\partial }{\partial x}(x - y) + (x - y)\frac{\partial }{\partial x}(x + y)\)

    \(\qquad = (x + y)(1) + (x - y)(1) = 2x\)

      And, \(f_{y} = 2y\)

3. Use the quotient rule to find the first-order partial derivatives for each of the following functions:

  1. \(f(x,y) = \frac{x + y}{2x}\)
  2. \(f(x,y) = \frac{x + y}{x - y}\)
  1. \(f_{x} = \frac{2x\frac{\partial }{\partial x}(x + y) - (x + y)\frac{\partial }{\partial x}(2x)}{(2x)^{2}} = \frac{2x(1) - (x + y)(2)}{4x^{2}} = -\frac{y}{2x^{2}}\)

      And, \(f_{y} = \frac{2x\frac{\partial }{\partial y}(x + y) - (x + y)\frac{\partial }{\partial y}(2x)}{(2x)^{2}} = \frac{2x(1) - (x + y)(0)}{4x^{2}} = \frac{1}{2x}\)

  2. \(f_{x} = \frac{(x - y)\frac{\partial }{\partial x}(x + y) - (x + y)\frac{\partial }{\partial x}(x - y)}{(x - y)^{2}} = \frac{(x - y)(1) - (x + y)(1)}{(x - y)^{2}} = -\frac{2y}{(x - y)^{2}}\)

      And, \(f_{y} = \frac{(x - y)\frac{\partial }{\partial y}(x + y) - (x + y)\frac{\partial }{\partial y}(x - y)}{(x - y)^{2}} = \frac{(x - y)(1) - (x + y)(-1)}{(x - y)^{2}} = \frac{2x}{(x - y)^{2}}\)


4. Find the first-order partial derivatives for each of the following functions:

  1. \(f(x,y) = x^{0.5}y^{0.5}\)
  2. \(f(x,y) = x^{0.6}y^{0.4}\)
  3. \(f(x,y) = 10x^{0.6}y^{0.4}\)
  4. \(f(x,y) = Ax^{\alpha }y^{\beta }\)
  1. \(f_{x} = 0.5x^{0.5 - 1}y^{0.5} = 0.5x^{-0.5}y^{0.5} = 0.5\frac{y^{0.5}}{x^{0.5}}\)

      And, \(f_{y} = 0.5x^{0.5}y^{0.5 - 1} = 0.5x^{0.5}y^{-0.5} = 0.5\frac{x^{0.5}}{y^{0.5}}\)

  2. \(f_{x} = 0.6x^{0.6 - 1}y^{0.4} = 0.6x^{-0.4}y^{0.4} = 0.6\frac{y^{0.4}}{x^{0.4}}\)

      And, \(f_{y} = 0.4x^{0.6}y^{0.4 - 1} = 0.4x^{0.6}y^{-0.6} = 0.4\frac{x^{0.6}}{y^{0.6}}\)

  3. \(f_{x} = 10(0.6)x^{0.6 - 1}y^{0.4} = 6x^{-0.4}y^{0.4} = 6\frac{y^{0.4}}{x^{0.4}}\)

      And, \(f_{y} = 10(0.4)x^{0.6}y^{0.4 - 1} = 4x^{0.6}y^{-0.6} = 4\frac{x^{0.6}}{y^{0.6}}\)

  4. \(f_{x} = A\alpha x^{\alpha - 1}y^{\beta} \qquad\qquad f_{y} = A\beta x^{\alpha }y^{\beta - 1}\)


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